1. Fi8000 Risk, Return and Portfolio Theory
Milind Shrikhande

2. Stock-Trak and Financial Links
See references in the book!
Yahoo! Finance:
Bloomberg:
Reuters:
CNBC:
CNNfn:
Wall Street Journal:
Morningstar:
CBOE:
CME:
NYSE:
NASDAQ:
AMEX:

3. Risk, Return and Portfolio Theory
Risk and risk aversion
Utility theory and the definition of risk aversion
Mean-Variance (M-V or μ-σ) criterion
The mathematics of portfolio theory
Capital allocation and the optimal portfolio
One risky asset and one risk-free asset
Two risky assets
n risky assets
n risky assets and one risk-free asset
Equilibrium in capital markets
The Capital Asset Pricing Model (CAPM)
Market Efficiency

4. Reward and Risk: Assumptions
Investors prefer more money (reward) to less: all else equal, investors prefer a higher reward to a lower one.
Investors are risk averse: all else equal, investors dislike risk.
There is a tradeoff between reward and risk: Investors will take risks only if they are compensated by a higher reward.

5. Reward and Risk
Reward ☺ ☺
Risk

6. Quantifying Rewards and Risks
A welfare measure
The expected (average) return
Risk
Measures of dispersion - variance
Other measures
The mathematics of portfolio theory (1-3)

7. Comparing Investments: an example
Which investment will you prefer and why?
A or B?
B or C?
C or D?
C or E?
D or E?
B or E, C or F?
E or F?

8. Comparing Investments: the criteria
A vs. B – If the return is certain look for the higher return (reward)
B vs. C – A certain dollar is always better than a lottery with an expected return of one dollar
C vs. D – If the expected return (reward) is the same look for the lower variance of the return (risk)
C vs. E – If the variance of the return (risk) is the same look for the higher expected return (reward)
D vs. E – Chose the investment with the lower variance of return (risk) and higher expected return (reward)
B vs. E or C vs. F – stochastic dominance
E vs. F – maximum expected utility

9. Comparing Investments
Maximum return
If the return is risk-free (certain), all investors prefer the higher return
Risk aversion
Investors prefer a certain dollar to a lottery with an expected return of one dollar

10. Comparing Investments
Maximum expected return
If two risky assets have the same variance of the returns, risk-averse investors prefer the one with the higher expected return*
Minimum variance of the return
If two risky assets have the same expected return, risk-averse investors prefer the one with the lower variance of return*

11. The Mean-Variance Criterion
Let A and B be two (risky) assets. All risk-averse investors prefer asset A to B if
{ μA ≥ μB and σA < σB }
or if
{ μA > μB and σA ≤ σB }
* Note that these rules apply only when we assume that the distribution of returns is normal.

12. The Utility of Certain Returns for Risk Averse Investors
Let us assume that the investor has an initial wealth of W0 dollars. Then
U(W0) < U(W0 + $1)
The utility is an increasing function of the final dollar wealth.
U(W0+$1) - U(W0) > U(W0+$101) - U(W0+$100)
The utility function is concave: the marginal utility is a decreasing function of the initial dollar wealth (an additional dollar has the same impact on the wealth of the “rich” investor, but it has a lower impact on the his welfare compared to the “poor” investor).

13. Risk-Averse Investors: Increasing and Concave Utility Function
U(W) W

14. The Expected Utility of Risky Returns
Let DA be the dollar return of asset A (a random variable), DA(i) be the dollar return of asset A in state i (i = 1, … n) and pi be the probability of state i. If you invest in A
Your expected dollar wealth is
E[W0+DA] = [W0+DA(1)]·p1 + … + [W0+DA(n)]·pn
Your expected utility (welfare) is
E[U(W0+DA)] = U[W0+DA(1)]·p1 + … + U[W0+DA(n)]·pn

15. Example (BKM page 193)
Consider a simple prospect where all your wealth of $100,000 is invested in a fair gamble: you will get $150,000 with probability 0.5 or $50,000 with probability 0.5.
Note that this is called a fair gamble since the expected profit is zero:
E(profit) = (150, ,000)·0.5 + (50, ,000) ·0.5
= 0.

16. Example - continued Calculate the expected final wealth. ($100,000)
Assuming that your utility function is logarithmic (i.e. U(W) = ln(W)), calculate your utility of the final wealth for each possible outcome. ( , )
Show that for this function the marginal utility is decreasing in the final wealth (numeric example).
Calculate the expected utility of the final wealth and compare it to the utility of the initial wealth. Will you enter the game? (f: , i: )
How much will you pay me for the right to enter this game? Or should I pay you? ($13,397.5)

17. The Certainty Equivalent
The certainty equivalent (CE) determines the maximum dollar price an investor will pay for a risky asset with an uncertain dollar return D.
The expected utility of the investment in the risky asset is equal to that of the certainty equivalent. In our example:
E[U(W0+D)] = 0.5·U($50K) + 0.5·U($150K) =
= U[CE]
CE = $86,602.5
That means that you will not invest more than $86,602.5 in that game, or that you will enter the game only if I will pay you a risk premium: $100,000 - $86,602.5 = $13,397.5

18. Example - continued
Assume that you invest $100,000 today (t = 0) and the outcome is expected a year from now (t = 1). What is the present value of the expected future CF if the risk-free rate is 5%? ($82,478.6)
What is your personal risk-premium in terms of the dollar difference between a certain CF at time t = 1 and the expected CF of this investment at t = 1? ($13,397.5)
What is your personal risk-premium in terms of the required rate of return? (k = 21.24%, k-rf = 16.24%)

19. Other Criteria
The basic intuition is that we care about “bad” surprises rather than all surprises. In fact dispersion (variance) may be desirable if it means that we may encounter a “good” surprise.
When we assume that returns are normally distributed the expected-utility and the stochastic-dominance criteria result in the same ranking of investments as the mean-variance criterion.

20. Practice problems BKM Ch. 6: 1,13,14 BKM Ch. 6, Appendix B: 1
Mathematics of Portfolio Theory:
Read and practice parts 1-10.